308 14 The FTAP for Unbounded Stochastic Processes
If no confusion can arise to which process the feasibility condition refers,
then we will simply say that the weight function is feasible. The first item
in the definition requires thatwis big enough in order to allow sufficiently
many integrandsHsuch that bothHand−Harew-admissible. The second
item requireswto be not too big, and as we will see, this will avoid arbitrage
opportunities. It follows from Proposition 14.2.6 and the assumption that
Meσ= 0, that the existence of feasible weight functions is guaranteed. We
also not that for locally bounded processesS, a functionw≥1isfeasibleas
soon as there isQ∈MeσwithEQ[w]<∞.
We can now state the generalisations of the duality theorem mentioned
above.
Theorem 14.5.9.Ifwis a feasible weight function andgis a random variable
such thatg≥−wthen:
sup
Q∈Meσ
EQ[w]<∞EQ[g]=inf{α| there isHw-admissible andg≤α+(H·S)∞}.If the quantities are finite then the infimum is a minimum.
Remark 14.5.10.The reader can see that even in the case of locally bounded
processesSthe result yields more precise information. Indeed we restrict the
supremum to those measuresQ∈Meσsuch thatEQ[w]<∞.
For a feasible weight functionw,wedenotebyKwthe setKw={(H·S)∞|Hisw-admissible}.Definition 14.5.11.An elementg∈Kwis called maximal if h∈Kwand
h≥gimply thath=g.
The maximal elements in this set are then characterised as follows:Theorem 14.5.12.Ifw≥ 1 is a feasible weight function, ifHisw-admissible
and ifh=(H·S)∞, then the following are equivalent:
(1)his maximal
(2)there isQ∈Meσsuch thatEQ[w]<∞andEQ[h]=0
(3)there isQ∈Meσ such thatEQ[w]<∞andH·S is aQ-uniformly
integrable martingale.
In the proof of these results we will make frequent use of Theorem 15.D and
Corollary 15.4.11. These two results were proved for a slightly more restrictive
notion of admissibility, but the reader can go through the proofs and check
that the results remain valid for the present notion ofw-admissible integrands.
Indeed the lower boundH·S≥−wis only used to control the negative parts
of the possible jumps in the stochastic integral. This can also be achieved